Groups associated with dynamical systems generalizing
Higman-Thompson groups
Volodymyr Nekrashevych
August 19, 2013
Durham
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Partial self-coverings
Let M be a topological space, and let M1 ⊆ M. Let f : M1 −→ M be a
finite degree covering map.
Examples: z 7→ z 2 acting on the unit circle (M1 = M).
z 7→ z 2 − 1 for M1 = C \ {0, 1, −1} and M = C \ {0, −1}.
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Let f : M1 −→ M be as above. For a point t ∈ M, define the tree of
preimages
G
Tt =
f −n (t).
n≥0
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Let γ be a path from t1 to t2 . Lifting γ by iterations of f we get an
isomorphism Sγ : Tt1 −→ Tt2 .
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Choose a basepoint t ∈ M. Consider two maximal antichains A1 , A2 ⊂ Tt
of equal cardinalities. Choose a bijection α : A1 −→ A2 and a collection of
paths γa from a ∈ A1 to α(a) ∈ A2 . These paths define isomorphisms
Sγa : Ta −→ Tα(a) . The union of the maps Sγa is an “almost
automorphism” S{γa : a∈A1 } of the tree Tt .
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Groups Vf
The set of all such almost automorphisms of Tt is a group, if we identify
two almost automorphisms that agree on all but a finite number of
vertices. Denote it by Vf .
Similarly, we can consider the inverse semigroup Pf generated by finite
unions of maps of the form Sγ .
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For every z ∈ f −1 (t) choose a path γz from t to z. Let Sz = Sγz be the
corresponding element
Pof Pf . The elements Sz satisfy the Cuntz algebra
relations: Sz∗ Sz = 1, z∈f −1 (t) Sz Sz∗ = 1.
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A product Sz1 Sz2 · · · Szn is equal to Sγ for a path γ starting at t and
ending in a vertex of the nth level of Tt . We get an isomorphism between
the tree of finite words X ∗ over the alphabet X = f −1 (t) and Tt . Denote
Sz1 z2 ...zn = Sz1 Sz2 · · · Szn .
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Higman-Thompson subgroup
The tree X ∗ consists of finite words over X , where a word v is connected
to vx for x ∈ X ∗ . The transformations Sx are the creation operators:
Sx (v ) = xv . Their inverses are the annihilation operators: Sx∗ (xv ) = v .
If A1 , A2 ⊂ X ∗ are maximal
antichains, and α : A1 −→ A2 is a bijection,
P
then the union g = v ∈A1 Sα(v ) Sv∗ is an element of Vf . It acts on (finite
or infinite) words by the rule
g (vw ) = α(v )w .
The set of such elements is naturally isomorphic to the Higman-Thompson
group Gd,1 for d = deg f .
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Let γ be a path in M from v to u ∈ X ∗ . Let γv and γu be such that
Sv = Sγv and Su = Sγu . Then γ ′ = γu−1 γγv ∈ π1 (M, t), and
Sγ = Su Sγ ′ Sv∗ . It follows that Vf is generated by Gn,1 and an image of
π1 (M, t).
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Examples
Let f be the map x 7→ 2x acting on the circle R/Z. Then Vf is generated
by a copy of V = G2,1 and a copy of Z = π1 (R/Z).
Choose the basepoint t = 0. Then f −1 (t) = {0, 1/2}. Let S0 be the trivial
path at 0, and let S1 be the image of the interval [0, 1/2]. Let a be the
generator of π1 (R/Z, 0) equal to the image of [0, 1].
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aS0 = S1 , aS1 = S0 a, a = S1 S0∗ + S0 aS1∗ .
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If f is z 7→ z 2 − 1 seen as a map C \ {0, ±1} −→ C \ {0, −1}, then Vf is
generated by V and two elements a, b satisfying
a = S1 S0∗ + S0 bS1∗ ,
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b = S0 S0∗ + S1 aS1∗ .
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Iterated monodromy group
The subgroup of Vf generated by π1 (M, t) is called the iterated
monodromy group of f , denoted IMG (f ).
Typically, it is infinitely presented, and often has exotic properties (e.g.,
intermediate growth, non-elementary amenable).
For example, IMG z 2 − 1 is the first example of an amenable group that
can not be constructed from groups of sub-exponential growth
(R. Grigorchuk-A.Žuk, L. Bartholdi-B. Virag). IMG z 2 + i has
intermediate growth (K.-U. Bux, R. Perez).
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For every g ∈ IMG (f ) and x ∈ X there exist unique y ∈ X and
h ∈ IMG (f ) such that gSx = Sy h. We denote y = g (x) and h = g |x .
Then
X
g=
Sg (x) g |x Sx∗ .
x∈X
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Definition
A group G of automorphisms of the tree X ∗ is said to be self-similar if for
every x ∈ X there exists h ∈ G such that
g (xw ) = g (x)h(w )
for all w ∈ X ∗ .
Let G be a self-similar group acting on X ∗ . Denote by VG the group
generated by G and the Higman-Thompson group G|X |,1 naturally almost
acting on X ∗ .
Theorem (C. Röver, 2002)
Let G be the Grigorchuk group. Then VG is isomorphic to the
commensurizer of G , is finitely presented, and simple.
For every self-similar group G , the group VG has simple commutator
subgroup.
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Expanding maps
The map f : M1 −→ M is said to be expanding (or hyperbolic) if there
exists a compact set J ⊂ M1 (its Julia set) such that f (J) = f −1 (J) = J,
and a metric d on J such that f is uniformly locally expanding on J.
(There exist ǫ, C > 0 and L > 1 such that
d(f −n (x), f −n (y )) ≤ CL−n d(x, y ) for all x, y such that d(x, y ) < ǫ.)
If f is hyperbolic, then IMG (f ) is contracting, i.e., there exists a finite set
N ⊂P
G such that for every element g ∈ IMG (f ) there exists n0 such that
g = v ∈X n Sg (v ) g |v Sv∗ for g |v ∈ N for all n ≥ n0 .
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Theorem
Suppose that G is a contracting self-similar group. Then VG is finitely
presented. In particular, if f is hyperbolic, then Vf is finitely presented.
Open questions: are contracting groups amenable? It is known that
IMG (f ), for f a post-critically finite polynomial, are amenable
(L. Bartholdi, V. Kaimanovich, V.N.).
When are contracting groups finitely presented? (Conjecturally, only when
they are virtually nilpotent.)
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Theorem
Let f1 , f2 be hyperbolic maps with connected Julia sets. Then Vf1 and Vf2
are isomorphic if and only if (f1 , J1 ) and (f2 , J2 ) are topologically
conjugate.
Ideas of the proof: It follows from M. Rubin’s theorem that every
isomorphism between Vfi comes from a homeomorphism between the
boundaries X ω of the corresponding trees. Action of Vfi on X ω generates
a hyperbolic groupoid. In particular, the graphs of the action on Vfi on
generic orbits are Gromov-hyperbolic. Their boundaries are one-point
compactification of an infinite “zoom” of Ji . A naturally defined dual
groupoid acting on the boundary is equivalent to the groupoid generated
by fi : Ji −→ Ji .
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Rabbit and Airplane
Consider the groups Vf associated with complex quadratic polynomials
whose critical point belongs to a cycle of length three. There exists two
such groups. Rabbit:
a = S1 S0∗ + S0 bS1∗ ,
b = S0 S0∗ + S1 cS1∗ ,
c = S0 S0∗ + S1 aS1∗ .
b = S0 S0∗ + S1 cS1∗ ,
c = S0 aS0∗ + S1 S1∗ .
Airplane:
a = S1 S0∗ + S0 bS1∗ ,
They are not isomorphic, since the corresponding Julia sets are not
homeomorphic.
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Cuntz-Pimsner algebras
C ∗ -algebras Of of operators on a Hilbert space generated by operators Sx ,
unitaries
g ∈ IMG (f ), subject to Cuntz algebra relations Sx∗ Sx = 1,
P
∗
x∈X Sx Sx = 1, and relations of the form gSx = Sy h, are examples of
Cuntz-Pimsner algebras.
Theorem
Let f be hyperbolic. Then Of is simple, nuclear (amenable), finitely
presented, and are classified by their K -theory (belong to the class of
Kirchberg-Phillips).
One can show, by computing the K -theory, that Of are pairwise
isomorphic for all hyperbolic complex quadratic polynomials f .
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